AI 中文总结
研究在\(\lambda>1\)为实代数数时，构造收敛半径为\(1\)的幂级数\(f(z)\)，其系数代数且模为\(\lambda^m\)，\(f^{(s)}\)在代数点取值代数且超越\(\mathbb{C}(z)\)，通过代数多边形消去与稀疏多项式块论证证明相关结论。
AI 中文摘要
设\(\lambda>1\)为实代数数。我们构造了连续统多个收敛半径恰好为\(1\)的幂级数\(f(z)=\sum_{k\geq0}a_kz^k\)，使得每个非零系数\(a_k\)都是代数的，且其模为\(\lambda^m\)（\(m\geq0\)）。此外，对于每个整数\(s\geq0\)，导数\(f^{(s)}\)在开单位圆盘的所有代数点处取代数值，并且在\(\mathbb{C}(z)\)上是超越的。证明结合了代数多边形消去法和稀疏多项式块论证。这表明一旦允许代数相位，系数模上的乘法秩一限制与每个代数点处全解析射流的代数性是兼容的。
英文摘要
Let $λ>1$ be a real algebraic number. We construct continuum many power series $f(z)=\sum_{k\geq0}a_kz^k$ of radius of convergence exactly one such that every nonzero coefficient $a_k$ is algebraic and has modulus $λ^m$ for some $m\geq0$. Moreover, for every integer $s\geq0$, the derivative $f^{(s)}$ takes algebraic values at all algebraic points of the open unit disk and is transcendental over $\mathbb{C}(z)$. The proof combines algebraic polygonal cancellation with a sparse polynomial-block argument. This shows that a multiplicative rank-one restriction on coefficient moduli is compatible with algebraicity of the full analytic jet at every algebraic point once algebraic phases are allowed.
